Axioms deciding the continuum hypothesis S arka Stejskalov a - - PowerPoint PPT Presentation

Axioms deciding the continuum hypothesis

ˇ S´ arka Stejskalov´ a

Kurt G¨

• del Research Center

University of Vienna logika.ff.cuni.cz/sarka

Prague May 2, 2018

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The continuum hypothesis

(CH) The continuum hypothesis says that any subset of the real numbers is at most countable or has the same size as the set of all real numbers. Or equivalently, the size of real numbers is the least possible, in a formula: 2ℵ0 = ℵ1.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The continuum hypothesis

(CH) The continuum hypothesis says that any subset of the real numbers is at most countable or has the same size as the set of all real numbers. Or equivalently, the size of real numbers is the least possible, in a formula: 2ℵ0 = ℵ1. Formulated by G. Cantor and popularised by D. Hilbert in 1900 (problem 1 on his list of 23 problems for the new century).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The independence CH over ZFC

Suppose ZFC is consistent, then the following hold:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The independence CH over ZFC

Suppose ZFC is consistent, then the following hold: (G¨

• del) ZFC + CH is consistent. (The least inner model L,

1930’s)

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The independence CH over ZFC

Suppose ZFC is consistent, then the following hold: (G¨

• del) ZFC + CH is consistent. (The least inner model L,

1930’s) (Cohen) ZFC + ¬CH is consistent. (Forcing, 1960’s)

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What does it mean?

The independence of CH has been interpreted in many ways (not mutually exclusive):

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What does it mean?

The independence of CH has been interpreted in many ways (not mutually exclusive): A1 It means that formalising set theory is hopeless if it cannot answer such a simple question.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What does it mean?

The independence of CH has been interpreted in many ways (not mutually exclusive): A1 It means that formalising set theory is hopeless if it cannot answer such a simple question. A2 It means that ZFC is a weak theory which should be extended.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What does it mean?

The independence of CH has been interpreted in many ways (not mutually exclusive): A1 It means that formalising set theory is hopeless if it cannot answer such a simple question. A2 It means that ZFC is a weak theory which should be extended. A3 It means that CH is a wrong question. One should look for a structure behind CH which is more robust.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What does it mean?

The independence of CH has been interpreted in many ways (not mutually exclusive): A1 It means that formalising set theory is hopeless if it cannot answer such a simple question. A2 It means that ZFC is a weak theory which should be extended. A3 It means that CH is a wrong question. One should look for a structure behind CH which is more robust. Regarding A3, note that CH just postulates the existence of a bijection between 2ℵ0 (R) and ℵ1; this by itself does not seem to be a structurally deep condition.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The generalized continuum hypothesis

We can ”globalize” the continuum hypothesis by postulating: (GCH) The generalised continuum hypothesis: 2ℵα = ℵα+1, for all α ∈ Ord.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The generalized continuum hypothesis

We can ”globalize” the continuum hypothesis by postulating: (GCH) The generalised continuum hypothesis: 2ℵα = ℵα+1, for all α ∈ Ord. Suppose ZFC is consistent, then the following hold:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The generalized continuum hypothesis

We can ”globalize” the continuum hypothesis by postulating: (GCH) The generalised continuum hypothesis: 2ℵα = ℵα+1, for all α ∈ Ord. Suppose ZFC is consistent, then the following hold: (G¨

• del) ZFC + GCH is consistent. (The least inner model L,

1930’s)

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The generalized continuum hypothesis

We can ”globalize” the continuum hypothesis by postulating: (GCH) The generalised continuum hypothesis: 2ℵα = ℵα+1, for all α ∈ Ord. Suppose ZFC is consistent, then the following hold: (G¨

• del) ZFC + GCH is consistent. (The least inner model L,

1930’s) (Cohen) ZFC + ¬GCH is consistent. (Forcing, 1960’s)

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The continuum function

Notice that the negation of GCH is not very informative: it just claims that there exists α with 2ℵα > ℵα+1. There are certainly more interesting “negations” of the GCH such as 2ℵα > ℵα+1 for all α. (We will discuss later whether this strong negation is consistent).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The continuum function

Notice that the negation of GCH is not very informative: it just claims that there exists α with 2ℵα > ℵα+1. There are certainly more interesting “negations” of the GCH such as 2ℵα > ℵα+1 for all α. (We will discuss later whether this strong negation is consistent). Even more generally, let us consider the function which maps infinite cardinals κ to 2κ. We call this function the continuum function.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The continuum function

Notice that the negation of GCH is not very informative: it just claims that there exists α with 2ℵα > ℵα+1. There are certainly more interesting “negations” of the GCH such as 2ℵα > ℵα+1 for all α. (We will discuss later whether this strong negation is consistent). Even more generally, let us consider the function which maps infinite cardinals κ to 2κ. We call this function the continuum function. To investigate failures of GCH, we study the patterns of the continuum function which are consistent with ZFC.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The continuum function

The following is provable about the continuum function in ZFC.

1 If κ < λ, then 2κ ≤ 2λ, 2 cf(2κ) > κ (in particular 2κ > κ). ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The continuum function

The following is provable about the continuum function in ZFC.

1 If κ < λ, then 2κ ≤ 2λ, 2 cf(2κ) > κ (in particular 2κ > κ).

As it turns out there is a big difference between the continuum on regular and singular cardinals. (A fact which is certainly not

• bvious and came as a surprise.)

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Regular cardinals

By Easton theorem (1970), ZFC has little control over the continuum function on regular cardinals. In fact, the continuum function (on regulars) only needs to satisfy the previous two conditions.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Regular cardinals

By Easton theorem (1970), ZFC has little control over the continuum function on regular cardinals. In fact, the continuum function (on regulars) only needs to satisfy the previous two conditions. In particular it is consistent that 2ℵα > ℵα+1 for all regular cardinals ℵα.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Singular cardinals

Singular cardinals do not allow such freedom: By the following theorem, GCH cannot fail first at strong limit singular cardinal with an uncountable cofinality.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Singular cardinals

Singular cardinals do not allow such freedom: By the following theorem, GCH cannot fail first at strong limit singular cardinal with an uncountable cofinality. (Silver 1975) If κ is a singular cardinal with an uncountable cofinality and GCH holds on a stationary set below κ then 2κ = κ+.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Singular cardinals

Singular cardinals do not allow such freedom: By the following theorem, GCH cannot fail first at strong limit singular cardinal with an uncountable cofinality. (Silver 1975) If κ is a singular cardinal with an uncountable cofinality and GCH holds on a stationary set below κ then 2κ = κ+. Note that as GCH holds below κ unboundedly often, κ is strong limit.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Singular cardinals

Singular cardinals do not allow such freedom: By the following theorem, GCH cannot fail first at strong limit singular cardinal with an uncountable cofinality. (Silver 1975) If κ is a singular cardinal with an uncountable cofinality and GCH holds on a stationary set below κ then 2κ = κ+. Note that as GCH holds below κ unboundedly often, κ is strong limit. Note that the assumption of uncountable cofinality is necessary as it is consistent that ℵω, the smallest singular cardinal, is the first cardinal where GCH fails. (Magidor 1977)

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

A strong failure of GCH

Recall that we mentioned the following strong failure of GCH: For all α, 2ℵα > ℵα+1.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

A strong failure of GCH

Recall that we mentioned the following strong failure of GCH: For all α, 2ℵα > ℵα+1. This statement also includes singular cardinals, so it is wise to expect difficulties in obtaining a model where this statement holds (due to the specific properties of singular cardinals). However, with the necessary use of large cardinals, Foreman and Woodin (1991) found a model of ZFC where this statement is true.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The smallest singular cardinal ℵω

There is the famous bound on 2ℵω identified by Shelah that for a strong limit ℵω: 2ℵω ≤ min(ℵ(2ω)+, ℵω4).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The smallest singular cardinal ℵω

There is the famous bound on 2ℵω identified by Shelah that for a strong limit ℵω: 2ℵω ≤ min(ℵ(2ω)+, ℵω4). The following is consistent:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The smallest singular cardinal ℵω

There is the famous bound on 2ℵω identified by Shelah that for a strong limit ℵω: 2ℵω ≤ min(ℵ(2ω)+, ℵω4). The following is consistent: 2ℵω = ℵω+n, 1 < n < ω (Magidor 1977).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The smallest singular cardinal ℵω

There is the famous bound on 2ℵω identified by Shelah that for a strong limit ℵω: 2ℵω ≤ min(ℵ(2ω)+, ℵω4). The following is consistent: 2ℵω = ℵω+n, 1 < n < ω (Magidor 1977). 2ℵω = ℵω+ω+1 (Magidor 1977), and 2ℵω = ℵω+α+1 for any ω ≤ α < ω1 (Shelah 1983).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The smallest singular cardinal ℵω

There is the famous bound on 2ℵω identified by Shelah that for a strong limit ℵω: 2ℵω ≤ min(ℵ(2ω)+, ℵω4). The following is consistent: 2ℵω = ℵω+n, 1 < n < ω (Magidor 1977). 2ℵω = ℵω+ω+1 (Magidor 1977), and 2ℵω = ℵω+α+1 for any ω ≤ α < ω1 (Shelah 1983). It is open whether 2ℵω can be greater or equal to ℵω1 (pcf conjecture implies no).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What should a new axiom ϕ deciding CH satisfy?

Classification using the concept of an intrinsic vs extrinsic axiom ϕ:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What should a new axiom ϕ deciding CH satisfy?

Classification using the concept of an intrinsic vs extrinsic axiom ϕ:

ϕ is intrinsic if it is contained in the iterative concept of set (all axioms ZFC are – arguably for AC – intrinsic).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What should a new axiom ϕ deciding CH satisfy?

Classification using the concept of an intrinsic vs extrinsic axiom ϕ:

ϕ is intrinsic if it is contained in the iterative concept of set (all axioms ZFC are – arguably for AC – intrinsic). ϕ is extrinsic if its used as an additional axiom based on external (practical) considerations such as its success rate at deciding new sentences or its potential to simplify arguments. GCH, ♦κ or κ are considered extrinsic (arguably MA, PFA).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What should a new axiom ϕ deciding CH satisfy?

Classification using the concept of an intrinsic vs extrinsic axiom ϕ:

ϕ is intrinsic if it is contained in the iterative concept of set (all axioms ZFC are – arguably for AC – intrinsic). ϕ is extrinsic if its used as an additional axiom based on external (practical) considerations such as its success rate at deciding new sentences or its potential to simplify arguments. GCH, ♦κ or κ are considered extrinsic (arguably MA, PFA).

ϕ should have deep structural consequences and create a beautiful theory (Shelah’s notion of a semi-axiom, sandwiched between “intrinsic” and “extrinsic”).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

What should a new axiom ϕ deciding CH satisfy?

Classification using the concept of an intrinsic vs extrinsic axiom ϕ:

ϕ is intrinsic if it is contained in the iterative concept of set (all axioms ZFC are – arguably for AC – intrinsic). ϕ is extrinsic if its used as an additional axiom based on external (practical) considerations such as its success rate at deciding new sentences or its potential to simplify arguments. GCH, ♦κ or κ are considered extrinsic (arguably MA, PFA).

ϕ should have deep structural consequences and create a beautiful theory (Shelah’s notion of a semi-axiom, sandwiched between “intrinsic” and “extrinsic”). Based on these criteria, ZFC + CH (or ZFC + 2ℵ0 = ℵ2) are not very appealing extensions of ZFC deciding CH.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Large cardinals

Many set-theoreticians, for instance G¨

• del and Woodin, view large

cardinals as being intrinsically motivated (recall that large-cardinal axioms strictly increase the consistency strength of ZFC).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Large cardinals

Many set-theoreticians, for instance G¨

• del and Woodin, view large

cardinals as being intrinsically motivated (recall that large-cardinal axioms strictly increase the consistency strength of ZFC). (G¨

• del). In 1950’s conjectured that large cardinals decide CH.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Large cardinals

Many set-theoreticians, for instance G¨

• del and Woodin, view large

cardinals as being intrinsically motivated (recall that large-cardinal axioms strictly increase the consistency strength of ZFC). (G¨

• del). In 1950’s conjectured that large cardinals decide CH.

(Solovay). In 1960’s showed that measurable cardinals (and many other) do not directly decide CH.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Large cardinals

Many set-theoreticians, for instance G¨

• del and Woodin, view large

cardinals as being intrinsically motivated (recall that large-cardinal axioms strictly increase the consistency strength of ZFC). (G¨

• del). In 1950’s conjectured that large cardinals decide CH.

(Solovay). In 1960’s showed that measurable cardinals (and many other) do not directly decide CH. (Woodin). Large cardinals – indirectly (in the background) – allow us to formulate extensions of ZFC which do decide CH.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Large cardinals in disguise

Large cardinals do not decide CH directly, but for instance ℵ2 may inherit some large-cardinal properties, which do decide CH.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Large cardinals in disguise

Large cardinals do not decide CH directly, but for instance ℵ2 may inherit some large-cardinal properties, which do decide CH. We say that κ is a former large cardinal if there is an inner model M ⊆ V and κ is a large cardinal in M (and typically stops being large in V ).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Compactness properties below ℵω

Recall that an uncountable regular cardinal κ has the tree property (TP(κ)), if every κ-tree has a cofinal branch. The intuition is that if TP(κ), then κ is compact in a similar sense as ω is.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Compactness properties below ℵω

Recall that an uncountable regular cardinal κ has the tree property (TP(κ)), if every κ-tree has a cofinal branch. The intuition is that if TP(κ), then κ is compact in a similar sense as ω is. Note that TP(κ) implies that κ is a former large cardinal (for instance TP(ℵ2) implies that ℵ2 is weakly compact in L).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Compactness properties below ℵω

Recall that an uncountable regular cardinal κ has the tree property (TP(κ)), if every κ-tree has a cofinal branch. The intuition is that if TP(κ), then κ is compact in a similar sense as ω is. Note that TP(κ) implies that κ is a former large cardinal (for instance TP(ℵ2) implies that ℵ2 is weakly compact in L). A natural question is whether TP(κ+), where κ is regular, has

• ther consequences on the continuum function beyond implying

κ<κ > κ. In particular TP(κ++) implies 2κ > κ+. We may consider this question locally for one cardinal, or globally with the notion of the continuum function.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

Compactness properties below ℵω

Recall that an uncountable regular cardinal κ has the tree property (TP(κ)), if every κ-tree has a cofinal branch. The intuition is that if TP(κ), then κ is compact in a similar sense as ω is. Note that TP(κ) implies that κ is a former large cardinal (for instance TP(ℵ2) implies that ℵ2 is weakly compact in L). A natural question is whether TP(κ+), where κ is regular, has

• ther consequences on the continuum function beyond implying

κ<κ > κ. In particular TP(κ++) implies 2κ > κ+. We may consider this question locally for one cardinal, or globally with the notion of the continuum function. Original results in this direction indicate that this is not the case:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property below ℵω

Theorem (Honzik, S. (2017)) Assume GCH. Suppose there are infinitely many weakly compact cardinals and let f : ω → ω be a monotonic function satisfying f (2n) ≥ 2n + 2, n < ω. Then there is a generic extension V [G] where the tree property holds at each ℵ2n, 0 < n < ω, and f determines the continuum function in V [G] below ℵω: 2ℵn = ℵf (n).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property at ℵω+2: finite gap

Note that with strong limit ℵω, TP(ℵω+2) implies 2ℵω > ℵω+1 which requires large cardinals. Yet the tree property does not pose any further restrictions on the value of 2ℵω, at least for a finite gap:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property at ℵω+2: finite gap

Note that with strong limit ℵω, TP(ℵω+2) implies 2ℵω > ℵω+1 which requires large cardinals. Yet the tree property does not pose any further restrictions on the value of 2ℵω, at least for a finite gap: Theorem (Friedman, Honzik, S. (2018)) Assume GCH. Suppose 0 ≤ n < ω, and κ is H(λ+n)-strong, where λ > κ is the least weakly compact above κ (with n = 0 we in addition require that the target model computes that λ is the least weakly compact cardinal above κ). Then there is a forcing extension where the following hold:

1 κ = ℵω is strong limit and 2ℵω = ℵω+2+n. 2 TP(ℵω+2). ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property at ℵω+2: infinite gap

The following important question is open for the moment: Is it consistent to have the following:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property at ℵω+2: infinite gap

The following important question is open for the moment: Is it consistent to have the following:

1 κ = ℵω is strong limit and 2ℵω = ℵα+1, for some α ≥ ω. 2 TP(ℵω+2). ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property at ℵω+2: infinite gap

The following important question is open for the moment: Is it consistent to have the following:

1 κ = ℵω is strong limit and 2ℵω = ℵα+1, for some α ≥ ω. 2 TP(ℵω+2).

While we expect it is consistent, singular cardinals do sometimes exhibit surprising behaviour, so the answer is far from clear.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property as a candidate for an extension of ZFC

Recall our discussion of theories extending ZFC which are supposed to “naturally” decide statements independent over ZFC, and consider the following theories:

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property as a candidate for an extension of ZFC

Recall our discussion of theories extending ZFC which are supposed to “naturally” decide statements independent over ZFC, and consider the following theories: “ZFC + TP(ℵ2)” is probably not a very good extension.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property as a candidate for an extension of ZFC

Recall our discussion of theories extending ZFC which are supposed to “naturally” decide statements independent over ZFC, and consider the following theories: “ZFC + TP(ℵ2)” is probably not a very good extension. “ZFC + TP(ℵn) for all 1 < n < ω” is a better extension, but still not good enough.

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property as a candidate for an extension of ZFC

Recall our discussion of theories extending ZFC which are supposed to “naturally” decide statements independent over ZFC, and consider the following theories: “ZFC + TP(ℵ2)” is probably not a very good extension. “ZFC + TP(ℵn) for all 1 < n < ω” is a better extension, but still not good enough. Smax = “ZFC + TP(κ) for all ℵ1 < κ, κ regular”, is more appealing (it is a major open problem whether this theory is consistent).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis

The tree property as a candidate for an extension of ZFC

Recall our discussion of theories extending ZFC which are supposed to “naturally” decide statements independent over ZFC, and consider the following theories: “ZFC + TP(ℵ2)” is probably not a very good extension. “ZFC + TP(ℵn) for all 1 < n < ω” is a better extension, but still not good enough. Smax = “ZFC + TP(κ) for all ℵ1 < κ, κ regular”, is more appealing (it is a major open problem whether this theory is consistent). A model of Smax is maximal in a strong sense: every regular cardinal larger than ℵ1 is a former large cardinal, and there are no counterexamples (Aronszajn trees) to compactness (except for ℵ1 which is provably incompact in ZFC).

ˇ

• S. Stejskalov´

a Axioms deciding the continuum hypothesis